IV. DISPLAYING CATEGORICAL DATA  

Introduction

A picture is said to be worth a thousand words.  Tables and graphs, properly designed, can provide clear pictures of patterns contained in many thousands of pieces of information.   In this topic, we will describe several ways of displaying information about categorical variables in tabular and graphic form.  In later topics, ways of displaying information about continuous variables will be explained.

Frequency Tables

A frequency table (or frequency distribution) displays numbers and percentages for each value of a variable.  It is useful for categorical variables (that is, those with values falling into a relatively small number of discrete categories, such as party identification, religious affiliation, or region of a country) rather than for continuous variables (such as age in years or gross domestic product in dollars). 

The following frequency distribution shows the number of seats in the U.S. House of Representatives by region of the country following the 2000 Census:

The first column in the table provides a label for each category of the variable.  The second and third columns show, respectively, the number and percent of cases in each category for all cases.  The fourth column shows the percent in each category after eliminating cases for which we do not have information (missing data).  Since we know the location of every congressional district, the fourth column is identical to the third in this case.   The last column shows the cumulative percentages as one goes from the first to the last category.  Note that this last column makes sense only if the values of the variable can be meaningfully ranked.  In other words, cumulative frequencies assume at least ordinal level measurement.  The numbers in this column make no sense in this example, since it wouldn't be meaningful to say that 42.1 percent of seats "are located in the Midwest or less." 

Contingency Tables

A contingency table (also called a crosstabulation, or crosstab for short) displays the relationship between one categorical variable and another.  It is called a “contingency table” because it allows us to examine a hypothesis that the values of one variable are contingent (dependent) upon those of another.

The following crosstabulation shows the relationship between region and party affiliation in the House of Representatives following the 2008 elections:

There are several important things to notice about the way in which the table has been set up:

• The table consists of a matrix of rows and columns.  Since there are two rows and four columns, it would be referred to as a “two by four” table.  A table with four rows and three columns would be referred to as a “four by three” table.  (Note that the number of rows is always given first.)  The values of one of the variables are placed in the rows and those of the other in the columns.   Usually, as in this case, the dependent variable is the row variable and the independent variable is the column variable.  There is, however, no requirement that this be the case.  If it fits the available space better to put the dependent variable in the columns, go ahead and do so.
   
• The number of cases in each cell of the table has been converted to a percent.  In this example Democrats dominated in the Northeast and, to a lesser extent, in the West; the Midwest and South were more evenly divided.  By converting numbers of cases to percents, we facilitate making comparisons among regions, despite fairly large differences in the number of seats each holds, since each totals to the same 100%.
   
• In this instance, we have percentaged down the columns rather than across the rows.  In using contingency tables to test hypotheses, always percentage in the direction of the independent variable.  This is necessary because we are testing the idea that different categories of the independent variable will tend to have different values for the dependent variable
  
  (On occasion, crosstabs are used, not to test hypotheses, but for purely descriptive purposes, and this general rule does not apply. For example, the 2008 American National Election Study Subset [see codebook] includes several variables thought to measure political efficacy [a person's belief that he or she can have an impact on politics]. If the variables included in the study are really measuring the same underlying concept, there should be a relationship between answers to one question and those to another, but we are not hypothesizing that either one depends on the other. In trying to see whether this is the case, we may decide to look, for each cell in the table, at the percent of the total table rather than of either the row or column. For another example of this, see the section on validity and reliability in the topic on "more about measurement.")
   
• The table includes information on the number of cases (N) on which the percentages are based.   When there are a large number of cases in each category, we can be more confident that the patterns we observe aren't merely coincidental.[1] 

Do not let all the trees get in the way of seeing the forest.  In interpreting a crosstab, it is crucial to focus on the overall picture.  In this case, the table shows that there are substantial regional differences in party strength.  Don’t get bogged down in the details.